In this talk, I will discuss progress in our understanding of Legendrian surfaces. First, I will present a new construction of Legendrian surfaces and a direct description for their moduli space of microlocal sheaves. This Legendrian invariant will connect to classical incidence problems in algebraic geometry and the study of flag varieties, which we will study in detail. There will be several examples during the talk and, in the end, I will indicate the relation of this theory to the study of framed local systems on a surface. This talk is based on my work with E. Zaslow.
One of the classical problems in scissors congruence is
this: given two polytopes in $n$-dimensional Euclidean space, when is
it possible to decompose them into finitely many pieces which are
pairwise congruent via translations? A complete set of invariants is
provided by the Hadwiger invariants, which measure "how much area is
pointing in each direction." Proving that these give a complete set
of invariants is relatively straightforward, but determining the
relations between them is much more difficult. This was done by
Dupont, in a 1982 paper. Unfortunately, this result is difficult to
describe and work with: it uses group homological techniques which
produce a highly opaque formula involving twisted coefficients and
relations in terms of uncountable sums. In this talk we will discuss
a new perspective on Dupont's proof which, together with more
topological simplicial techniques, simplifies and clarifies the
classical results. This talk is partially intended to be an
advertisement for simplicial techniques, and will be suitable for
graduate students and others unfamiliar with the approach.
In network routing users often tradeoff different objectives in selecting their best route. An example is transportation networks, where due to uncertainty of travel times, drivers may tradeoff the average travel time versus the variance of a route. Or they might tradeoff time and cost, such as the cost paid in tolls.
We wish to understand the effect of two conflicting criteria in route selection, by studying the resulting traffic assignment (equilibrium) in the network. We investigate two perspectives of this topic: (1) How does the equilibrium cost of a risk-averse population compare to that of a risk-neutral population? (i.e., how much longer do we spend in traffic due to being risk-averse) (2) How does the equilibrium cost of a heterogeneous population compare to that of a comparable homogeneous user population?
We provide characterizations to both questions above.
Based on joint work with Richard Cole, Thanasis Lianeas and Nicolas Stier-Moses.
At the end I will mention current work of my research group on algorithms and mechanism design for power systems.
Biography: Evdokia Nikolova is an Assistant Professor in the Department of Electrical and Computer Engineering at the University of Texas at Austin, where she is a member of the Wireless Networking & Communications Group. Previously she was an Assistant Professor in Computer Science and Engineering at Texas A&M University. She graduated with a BA in Applied Mathematics with Economics from Harvard University, MS in Mathematics from Cambridge University, U.K. and Ph.D. in Computer Science from MIT.
Nikolova's research aims to improve the design and efficiency of complex systems (such as networks and electronic markets), by integrating stochastic, dynamic and economic analysis. Her recent work examines how human risk aversion transforms traditional computational models and solutions. One of her algorithms has been adapted in the MIT CarTel project for traffic-aware routing. She currently focuses on developing algorithms for risk mitigation in networks, with applications to transportation and energy. She is a recipient of an NSF CAREER award and a Google Faculty Research Award. Her research group has been recognized with a best student paper award and a best paper award runner-up. She currently serves on the editorial board of the journal Mathematics of Operations Research.
For a first order (deterministic) mean-field game with non-local running and initial couplings, a classical solution is constructed for the associated, so-called master equation, a partial differential equation in infinite-dimensional space with a non-local term, assuming the time horizon is sufficiently small and the coefficients are smooth enough, without convexity conditions on the Hamiltonian.
In independent bond percolation with parameter p, if one removes the vertices of the infinite cluster (and incident edges), for which values of p does the remaining graph contain an infinite cluster? Grimmett-Holroyd-Kozma used the triangle condition to show that for d > 18, the set of such p contains values strictly larger than the percolation threshold pc. With the work of Fitzner-van der Hofstad, this has been reduced to d > 10. We reprove this result by showing that for d > 10 and some p>pc, there are infinite paths consisting of "shielded"' vertices --- vertices all whose adjacent edges are closed --- which must be in the complement of the infinite cluster. Using numerical values of pc, this bound can be reduced to d > 7. Our methods are elementary and do not require the triangle condition.
Invasion percolation is a stochastic growth model that follows a greedy algorithm. After assigning i.i.d. uniform random variables (weights) to all edges of d-dimensional space, the growth starts at the origin. At each step, we adjoin to the current cluster the edge of minimal weight from its boundary. In '85, Chayes-Chayes-Newman studied the "acceptance profile"' of the invasion: for a given p in [0,1], it is the ratio of the expected number of invaded edges until time n with weight in [p,p+dp] to the expected number of observed edges (those in the cluster or its boundary) with weight in the same interval. They showed that in all dimensions, the acceptance profile an(p) converges to one for p<pc and to zero for p>pc. In this paper, we consider an(p) at the critical point p=pc in two dimensions and show that it is bounded away from zero and one as n goes to infinity.
An electron interacting with the vibrational modes of a polar crystal is called a polaron. Polarons are the simplest Quantum Field Theory models, yet their most basic features such as the effective mass, ground-state energy and wave function cannot be evaluated explicitly. And while several successful theories have been proposed over the years to approximate the energy and effective mass of various polarons, they are built entirely on unjustified, even questionable, Ansätze for the wave function.
In this talk I shall provide the first explicit description of the ground-state wave function of a polaron in an asymptotic regime: For the Fröhlich polaron localized in a Coulomb potential and exposed to a homogeneous magnetic field of strength $B$ it will be shown that the ground-state electron density in the direction of the magnetic field converges pointwise and in a weak sense as $B\rightarrow\infty$ to the square of a hyperbolic secant function--a sharp contrast to the Gaussian wave functions suggested in the physics literature.